Tetrakis Hexahedral Graph

Author

Eric W. Weisstein
September 16, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/TetrakisHexahedralGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TetrakisHexahedralGraph.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graph

GraphData["TetrakisHexahedralGraph","StandardName"]

Primary

GraphData["TetrakisHexahedralGraph"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["TetrakisHexahedralGraph","Graph","All"]//StyleGraphs

Default embedding

GraphPlot@GraphData["TetrakisHexahedralGraph","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["TetrakisHexahedralGraph","Graph","Degenerate"]//StyleGraphs

Circular

GraphData["TetrakisHexahedralGraph","Graph","Circular"]//StyleGraphs

Planar

GraphData["TetrakisHexahedralGraph","Planar"]

GraphData["TetrakisHexahedralGraph","Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["TetrakisHexahedralGraph"]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData["TetrakisHexahedralGraph"],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction 3

GraphPlot[RotateGraph[,-2Pi/3]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

LCF

GraphData["TetrakisHexahedralGraph","LCF"]

GraphData["TetrakisHexahedralGraph","Graph","LCF"]//StyleGraphs

Perspective

GraphData["TetrakisHexahedralGraph","Graph","Perspective"]//StyleGraphs

UnitDistance

GraphData["TetrakisHexahedralGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["TetrakisHexahedralGraph"],Debug->True]//Timing

GraphData["TetrakisHexahedralGraph","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["TetrakisHexahedralGraph","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData["TetrakisHexahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData["TetrakisHexahedralGraph","Matchstick"]

GraphData["TetrakisHexahedralGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["TetrakisHexahedralGraph","Edges"]]

3D All

GraphData["TetrakisHexahedralGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["TetrakisHexahedralGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["TetrakisHexahedralGraph","Polyhedral"]

GraphData["TetrakisHexahedralGraph","Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData["TetrakisHexahedralGraph","Polyhedron"]}]

3D Canonical

GraphData["TetrakisHexahedralGraph","Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData["TetrakisHexahedralGraph","Polyhedron","Canonical"],Yellow,Sphere[]}]

Embeddings

GraphDataEmbeddings[GraphData["TetrakisHexahedralGraph","EdgeList"]]

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["TetrakisHexahedron","Polyhedron"],#,"Simplification"->FullSimplify]&/@{{1,1,0},{(Sqrt[2]),0,1},{2(Sqrt[2]),0,1},{((2 (Sqrt[2]))/5),0,1},{1,1,(5/(Sqrt[2])),ArcTan[(5/(Sqrt[29]))]},{1,1,(1/(Sqrt[2])),ArcCot[(Sqrt[5])]}}

GraphData["TetrakisHexahedralGraph","Graph","Perspective"]

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["TetrakisHexahedralGraph"]]]

Properties

GraphData[g,"Missing"]

GraphData[g,{"V","M"}]

TableForm[GraphDataTable[g,x],TableDepth->2]//TraditionalForm

Embeddings by Projection

PolyhedronData[poly]

Degenerate

GraphPlot[SkeletonGraph[PolyhedronData[poly,"Faces"]//Normal,#],Method->None]&/@{1.9,2.7,3,3.5,5,6}

rs=RecognizeGraph/@(gs=ToCombinatoricaGraph/@{})

ToCommonEdges[#,rs[[1]]]&/@gs

GraphData

GraphClasses

GraphClassString[{g}]